Deflector beam in The Paradise Syndrome
Posted: Wed Jan 27, 2016 3:02 am
This is an analysis of the power of the Enterprise 1701's deflector beam - which will place a minimum limit upon the overall power generation capacity of 22nd century ships.
First, some relevant quotes from The Paradise Syndrome:
SPOCK: Prepare to beam us up, Mister Scott. We're warping out of orbit.
MCCOY: Leaving? You can't be serious, Spock.
SPOCK: Doctor, that asteroid is almost as large as your Earth's moon. Far enough away, the angle necessary to divert it enough to avoid destruction is minute, but as the asteroid approaches this planet, the angle becomes so great that even the power of a starship
MCCOY: The devil with an asteroid! It won't get here for two months, Spock!
SPOCK: If we arrive at the deflection point in time, it may not get here at all.
...
SPOCK: Prepare to activate deflectors.
SULU: Aye, sir.
CHEKOV: Power dropping, sir.
(The lights dim.)
SPOCK: Engineering, maintain full power. Full power.
SCOTT [OC]: Dilithium crystal circuit's failing, sir. We'll have to replace it.
SPOCK: Not now.
CHEKOV: Zero. Deflection point now, sir.
SPOCK: Activate deflectors.
(A beam shoots out and pulses against the big lump of rock.)
CHEKOV: Power dropping, sir.
SPOCK: Degree of deflection, Mister Sulu.
SULU: Not enough, Mister Spock. It's only point zero zero one three degrees.
...
SCOTT [OC]: Don't ask for anymore warp nine speeds, Mister Spock. Our star drive is completely burned out. The only thing we have left is impulse power.
SPOCK: Estimated repair time?
{Engineering]
SCOTT: Hanging here in space? Forever.
[Spock's quarters]
SCOTT [OC]: The only thing that'll fix these poor darlings is the nearest repair base.
SPOCK: I've already surmised that. Thank you, Mister Scott.
MCCOY: Well, Spock, you took your calculated risk in your calculated Vulcan way, and you lost. You lost for us, you lost for that planet, and you lost for Jim.
SPOCK: I accept the responsibility, Doctor.
MCCOY: And my responsibility is the health of this crew. You've been driving yourself too hard, and I want you to get some rest.
SPOCK: Mister Chekov, resume heading eight eight three mark four one.
MCCOY: Back to that planet? Without warp speed, it'll take months, Spock.
SPOCK: Exactly fifty nine point two two three days, Doctor, and that asteroid will be four hours behind us all the way.
So, the key data points are:
-the asteroid was almost as large as our moon. We may assume it has a mass of some 1e22 kg, which is a bit more than an eighth of the mass of Luna. That's pushing the meaning of the phrase "almost as large as," but it will make the math easier, and will give us a measure of conservatism.
-the asteroid was traveling almost as fast as the Enterprise at "maximum impulse." In Star Trek: The Motion Picture, the newly refit Enterprise got from earth to Jupiter in 1.8 hours. That's a minimum of some 97,000,000 m/s. Assuming that the original Enterprise was only capable of reaching half that speed, the asteroid is moving at a speed of some 48,000,000 m/s or more.
-The Enterprise deflected it by some 0.0013 degrees.
What we need to find is the Delta-V (change in velocity) of the asteroid.
Now the formula for this situation is:
Delta-V = 4.8e7 m/s x 2 x sin(0.5 x 0.0013 degrees)
= 4.8e7 m/s x 2 x 0.00001134454014
= 4.8e7 m/s x 0.00002268928028
= 1089 m/s
So the constitution managed to accelerate the asteroid by around a kilometer per second. Somehow, this wasn't enough to get it out of the way, but if the planetary system had a complicated enough arrangement over these two months, there could be some sort of unique gravitational funneling effect. At any rate, the total energy imparted to the asteroid was well in excess of 1e25 J. Which means that the power of the TOS Enterprise is well in excess of a hundred billion terawatts.
EDIT: For some reason, I just multiplied the mass of the moon by the delta-V, when what I should have done is multiply the mass of the moon by the square of the delta-V. So the actual amount of energy the Enterprise imparted to the asteroid would have been on the order of 1e28 J, which puts us in the vicinity of a hundred trillion terawatts.
First, some relevant quotes from The Paradise Syndrome:
SPOCK: Prepare to beam us up, Mister Scott. We're warping out of orbit.
MCCOY: Leaving? You can't be serious, Spock.
SPOCK: Doctor, that asteroid is almost as large as your Earth's moon. Far enough away, the angle necessary to divert it enough to avoid destruction is minute, but as the asteroid approaches this planet, the angle becomes so great that even the power of a starship
MCCOY: The devil with an asteroid! It won't get here for two months, Spock!
SPOCK: If we arrive at the deflection point in time, it may not get here at all.
...
SPOCK: Prepare to activate deflectors.
SULU: Aye, sir.
CHEKOV: Power dropping, sir.
(The lights dim.)
SPOCK: Engineering, maintain full power. Full power.
SCOTT [OC]: Dilithium crystal circuit's failing, sir. We'll have to replace it.
SPOCK: Not now.
CHEKOV: Zero. Deflection point now, sir.
SPOCK: Activate deflectors.
(A beam shoots out and pulses against the big lump of rock.)
CHEKOV: Power dropping, sir.
SPOCK: Degree of deflection, Mister Sulu.
SULU: Not enough, Mister Spock. It's only point zero zero one three degrees.
...
SCOTT [OC]: Don't ask for anymore warp nine speeds, Mister Spock. Our star drive is completely burned out. The only thing we have left is impulse power.
SPOCK: Estimated repair time?
{Engineering]
SCOTT: Hanging here in space? Forever.
[Spock's quarters]
SCOTT [OC]: The only thing that'll fix these poor darlings is the nearest repair base.
SPOCK: I've already surmised that. Thank you, Mister Scott.
MCCOY: Well, Spock, you took your calculated risk in your calculated Vulcan way, and you lost. You lost for us, you lost for that planet, and you lost for Jim.
SPOCK: I accept the responsibility, Doctor.
MCCOY: And my responsibility is the health of this crew. You've been driving yourself too hard, and I want you to get some rest.
SPOCK: Mister Chekov, resume heading eight eight three mark four one.
MCCOY: Back to that planet? Without warp speed, it'll take months, Spock.
SPOCK: Exactly fifty nine point two two three days, Doctor, and that asteroid will be four hours behind us all the way.
So, the key data points are:
-the asteroid was almost as large as our moon. We may assume it has a mass of some 1e22 kg, which is a bit more than an eighth of the mass of Luna. That's pushing the meaning of the phrase "almost as large as," but it will make the math easier, and will give us a measure of conservatism.
-the asteroid was traveling almost as fast as the Enterprise at "maximum impulse." In Star Trek: The Motion Picture, the newly refit Enterprise got from earth to Jupiter in 1.8 hours. That's a minimum of some 97,000,000 m/s. Assuming that the original Enterprise was only capable of reaching half that speed, the asteroid is moving at a speed of some 48,000,000 m/s or more.
-The Enterprise deflected it by some 0.0013 degrees.
What we need to find is the Delta-V (change in velocity) of the asteroid.
Now the formula for this situation is:
Delta-V = 4.8e7 m/s x 2 x sin(0.5 x 0.0013 degrees)
= 4.8e7 m/s x 2 x 0.00001134454014
= 4.8e7 m/s x 0.00002268928028
= 1089 m/s
So the constitution managed to accelerate the asteroid by around a kilometer per second. Somehow, this wasn't enough to get it out of the way, but if the planetary system had a complicated enough arrangement over these two months, there could be some sort of unique gravitational funneling effect. At any rate, the total energy imparted to the asteroid was well in excess of 1e25 J. Which means that the power of the TOS Enterprise is well in excess of a hundred billion terawatts.
EDIT: For some reason, I just multiplied the mass of the moon by the delta-V, when what I should have done is multiply the mass of the moon by the square of the delta-V. So the actual amount of energy the Enterprise imparted to the asteroid would have been on the order of 1e28 J, which puts us in the vicinity of a hundred trillion terawatts.